Let \alpha and \beta be the distinct roots of the equation x^2+x-1=0. Consider the set T=\{1, \alpha, \beta\}. For a 3 \times 3 matrix M= (a_{i j}) 3 \times 3), define R_i=a_{i 1}+a_{i 2}+a_\beta and C_j=a_{1 j}+a_{2 j}+a_{3 j} for i=1,2,3 and j=1,2,3 Match each entry in List-I to the correct entry in List-II. List-I List-II (P) The number of matrices M= (a_{i j}) 3 \times 3 with all entries in T such that R_i=C_j=0 for all i, j is (1) 1 (Q) The number of symmetric matrices M= (a_{i j}) 3 \times 3 with all entries in T such that C_j=0 for all j is (2) 12 (R) Let M= (a_{i j}) _{3 \times 3} be a skew symmetric matrix such that a_{ij} \in T for i>j . Then the number of elements in the set \{\left(\begin{array}{c}x \ y \ z)\end{array}: x, y \cdot z \in R, M\left(\begin{array}{c}x \ y \ z)\end{array}=\left(\begin{array}{c}a_{12\end{array} ; 0 ; -a_{23})}\} is (3) Infinite (S) Let M= (a_{i j}) _{3 \times 3} be a matrix with all entries in T such that R_i=0 for all i . Then the absolute value of the determinant of M is (4) 6 (5) 0 The correct option is [JEE Adv 2024 P1]

Correct Answer is : (P) ⟶ (2) (Q) ⟶ (4) (R) ⟶ (3) (S) ⟶ (5)

(P) \rightarrow (4) (Q) \rightarrow (2) (R) \rightarrow (5) (S) \rightarrow (1)

(P) \rightarrow(2)( {\Q}) \rightarrow(4)( {\R}) \rightarrow(1)( {\S}) \rightarrow(5)

(P) \rightarrow (2) (Q) \rightarrow (4) (R) \rightarrow (3) (S) \rightarrow (5)

(P) \rightarrow (1) (Q) \rightarrow (5) (R) \rightarrow (3) (S) \rightarrow (4)

Matrices and Determinants

Section: Mathematics

Question : 5 of 56

Marks: +1, -0

Let α and β be the distinct roots of the equation x2+x−1=0. Consider the set T={1,α,β}. For a 3×3 matrix M=(aij)3×3), define Ri=ai1+ai2+aβ and Cj=a1j+a2j+a3j for i=1,2,3 and j=1,2,3
Match each entry in List-I to the correct entry in List-II.

List-I

List-II

(P)

The number of matrices M=(aij)3×3 with all entries in T such that Ri=Cj=0 for all i,j is

(1)

(Q)

The number of symmetric matrices M=(aij)3×3 with all entries in T such that Cj=0 for all j is

(2)

(R)

Let M=(aij)3×3 be a skew symmetric matrix such that aij∈T for i>j . Then the number of elements in the set

{(

):x,y⋅z∈R,M(

)=(

a12

−a23

)} is

(3)

Infinite

(S)

Let M=(aij)3×3 be a matrix with all entries in T such that Ri=0 for all i . Then the absolute value of the determinant of M is

(4)

(5)

The correct option is

[JEE Adv 2024 P1]

(P) ⟶ (4) (Q) ⟶ (2) (R) ⟶ (5) (S) ⟶ (1)
(P) ⟶(2)(Q)⟶(4)(R)⟶(1)(S)⟶(5)
(P) ⟶ (2) (Q) ⟶ (4) (R) ⟶ (3) (S) ⟶ (5)
(P) ⟶ (1) (Q) ⟶ (5) (R) ⟶ (3) (S) ⟶ (4)

Validate

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